On b-colorings and b-continuity of graphs

A b-coloring of G is a proper vertex coloring such that there is a vertex in each color class, which is adjacent to at least one vertex in every other color class. Such a vertex is called a color-dominating vertex. The b-chromatic number of G is the largest k such that there is a b-coloring of G by k colors. Moreover, if for every integer k, between chromatic number and b-chromatic number, there exists a b-coloring of G by k colors, then G is b-continuous. Determining the b-chromatic number of a graph G and the decision whether the given graph G is b-continuous or not is NP-hard. Therefore, it is interesting to find new results on b-colorings and b-continuity for special graphs. In this thesis, for several graph classes some exact values as well as bounds of the b-chromatic number were ascertained. Among all we considered graphs whose independence number, clique number, or minimum degree is close to its order as well as bipartite graphs. The investigation of bipartite graphs was based on considering of the so-called bicomplement which is used to determine the b-chromatic number of special bipartite graphs, in particular those whose bicomplement has a simple structure. Then we studied some graphs whose b-chromatic number is close to its t-degree. At last, the b-continuity of some graphs is studied, for example, for graphs whose b-chromatic number was already established in this thesis. In particular, we could prove that Halin graphs are b-continuous.:Contents 1 Introduction 2 Preliminaries 2.1 Basic terminology 2.2 Colorings of graphs 2.2.1 Vertex colorings 2.2.2 a-colorings 3 b-colorings 3.1 General bounds on the b-chromatic number 3.2 Exact values of the b-chromatic number for special graphs 3.2.1 Graphs with maximum degree at most 2 3.2.2 Graphs with independence number close to its order 3.2.3 Graphs with minimum degree close to its order 3.2.4 Graphs G with independence number plus clique number at most number of vertices 3.2.5 Further known results for special graphs 3.3 Bipartite graphs 3.3.1 General bounds on the b-chromatic number for bipartite graphs 3.3.2 The bicomplement 3.3.3 Bicomplements with simple structure 3.4 Graphs with b-chromatic number close to its t-degree 3.4.1 Regular graphs 3.4.2 Trees and Cacti 3.4.3 Halin graphs 4 b-continuity 4.1 b-spectrum of special graphs 4.2 b-continuous graph classes 4.2.1 Known b-continuous graph classes 4.2.2 Halin graphs 4.3 Further graph properties concerning b-colorings 4.3.1 b-monotonicity 4.3.2 b-perfectness 5 Conclusion Bibliograph

On the oriented chromatic number of dense graphs

Let $G$ be a graph with $n$ vertices, $m$ edges, average degree $\delta$, and maximum degree $\Delta$. The \emph{oriented chromatic number} of $G$ is the maximum, taken over all orientations of $G$, of the minimum number of colours in a proper vertex colouring such that between every pair of colour classes all edges have the same orientation. We investigate the oriented chromatic number of graphs, such as the hypercube, for which $\delta\geq\log n$. We prove that every such graph has oriented chromatic number at least $\Omega(\sqrt{n})$. In the case that $\delta\geq(2+\epsilon)\log n$, this lower bound is improved to $\Omega(\sqrt{m})$. Through a simple connection with harmonious colourings, we prove a general upper bound of \Oh{\Delta\sqrt{n}} on the oriented chromatic number. Moreover this bound is best possible for certain graphs. These lower and upper bounds are particularly close when $G$ is ($c\log n$)-regular for some constant $c>2$, in which case the oriented chromatic number is between $\Omega(\sqrt{n\log n})$ and $\mathcal{O}(\sqrt{n}\log n)$

On the choosability of HH-minor-free graphs

Given a graph $H$, let us denote by $f_\chi(H)$ and $f_\ell(H)$, respectively, the maximum chromatic number and the maximum list chromatic number of $H$-minor-free graphs. Hadwiger's famous coloring conjecture from 1943 states that $f_\chi(K_t)=t-1$ for every $t \ge 2$. In contrast, for list coloring it is known that $2t-o(t) \le f_\ell(K_t) \le O(t (\log \log t)^6)$ and thus, $f_\ell(K_t)$ is bounded away from the conjectured value $t-1$ for $f_\chi(K_t)$ by at least a constant factor. The so-called $H$-Hadwiger's conjecture, proposed by Seymour, asks to prove that $f_\chi(H)=\textsf{v}(H)-1$ for a given graph $H$ (which would be implied by Hadwiger's conjecture). In this paper, we prove several new lower bounds on $f_\ell(H)$, thus exploring the limits of a list coloring extension of $H$-Hadwiger's conjecture. Our main results are: For every $\varepsilon>0$ and all sufficiently large graphs $H$ we have $f_\ell(H)\ge (1-\varepsilon)(\textsf{v}(H)+\kappa(H))$, where $\kappa(H)$ denotes the vertex-connectivity of $H$. For every $\varepsilon>0$ there exists $C=C(\varepsilon)>0$ such that asymptotically almost every $n$-vertex graph $H$ with $\left\lceil C n\log n\right\rceil$ edges satisfies $f_\ell(H)\ge (2-\varepsilon)n$. The first result generalizes recent results on complete and complete bipartite graphs and shows that the list chromatic number of $H$-minor-free graphs is separated from the natural lower bound $(\textsf{v}(H)-1)$ by a constant factor for all large graphs $H$ of linear connectivity. The second result tells us that even when $H$ is a very sparse graph (with an average degree just logarithmic in its order), $f_\ell(H)$ can still be separated from $(\textsf{v}(H)-1)$ by a constant factor arbitrarily close to $2$. Conceptually these results indicate that the graphs $H$ for which $f_\ell(H)$ is close to $(\textsf{v}(H)-1)$ are typically rather sparse.Comment: 14 page

Random Interval Graphs

In this thesis, which is supervised by Dr. David Penman, we examine random interval graphs. Recall that such a graph is defined by letting $X_{1},\ldots X_{n},Y_{1},\ldots Y_{n}$ be $2n$ independent random variables, with uniform distribution on $[0,1]$. We then say that the $i$th of the $n$ vertices is the interval $[X_{i},Y_{i}]$ if $X_{i}<Y_{i}$ and the interval $[Y_{i},X_{i}]$ if $Y_{i}<X_{i}$. We then say that two vertices are adjacent if and only if the corresponding intervals intersect. We recall from our MA902 essay that fact that in such a graph, each edge arises with probability $2/3$, and use this fact to obtain estimates of the number of edges. Next, we turn to how these edges are spread out, seeing that (for example) the range of degrees for the vertices is much larger than classically, by use of an interesting geometrical lemma. We further investigate the maximum degree, showing it is always very close to the maximum possible value $(n-1)$, and the striking result that it is equal to $(n-1)$ with probability exactly $2/3$. We also recall a result on the minimum degree, and contrast all these results with the much narrower range of values obtained in the alternative \lq comparable\rq\, model $G(n,2/3)$ (defined later). We then study clique numbers, chromatic numbers and independence numbers in the Random Interval Graphs, presenting (for example) a result on independence numbers which is proved by considering the largest chain in the associated interval order. Last, we make some brief remarks about other ways to define random interval graphs, and extensions of random interval graphs, including random dot product graphs and other ways to define random interval graphs. We also discuss some areas these ideas should be usable in. We close with a summary and some comments

Hipergráfok = Hypergraphs

A projekt célkitűzéseit sikerült megvalósítani. A négy év során több mint száz kiváló eredmény született, amiből eddig 84 dolgozat jelent meg a téma legkiválóbb folyóirataiban, mint Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, stb. Számos régóta fennálló sejtést bebizonyítottunk, egész régi nyitott problémát megoldottunk hipergráfokkal kapcsolatban illetve kapcsolódó területeken. A problémák némelyike sok éve, olykor több évtizede nyitott volt. Nem egy közvetlen kutatási eredmény, de szintén bizonyos értékmérő, hogy a résztvevők egyike a Norvég Királyi Akadémia tagja lett és elnyerte a Steele díjat. | We managed to reach the goals of the project. We achieved more than one hundred excellent results, 84 of them appeared already in the most prestigious journals of the subject, like Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, etc. We proved several long standing conjectures, solved quite old open problems in the area of hypergraphs and related subjects. Some of the problems were open for many years, sometimes for decades. It is not a direct research result but kind of an evaluation too that a member of the team became a member of the Norvegian Royal Academy and won Steele Prize